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Problem 952

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Order Modulo Factorial

Given a prime $p$ and a positive integer $n \lt p$, let $R(p, n)$ be the multiplicative order of $p$ modulo $n!$.

In other words, $R(p, n)$ is the minimal positive integer $r$ such that
$$p^r \equiv 1 \pmod{n!}$$
For example, $R(7, 4) = 2$ and $R(10^9 + 7, 12) = 17280$.

Find $R(10^9 + 7, 10^7)$. Give your answer modulo $10^9 + 7$.

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模阶乘的阶

给定质数$p$和正整数$n \lt p$，记$R(p, n)$为$p$模$n!$的阶。
换句话说，$R(p, n)$是满足下式的最小正整数$r$：
$$p^r \equiv 1 \pmod{n!}$$
例如，$R(7, 4) = 2$，$R(10^9 + 7, 12) = 17280$。

求$R(10^9 + 7, 10^7)$，并对$10^9 + 7$取余作为你的答案。

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